What causes these shadow artifacts?
My guess - The bevel modifier will try to bevel every edge, depending on the bevel setting... if the "limit" setting is set to "none" then it will try to bevel all edges, even the ones running through an area that should be flat. This can create unwanted "wrinkles" or as it tries to create indentations wherever you have extra edge loops vertices. 

For example, in the pic below, the right side of the cube has no edge loops so its bevel looks clean, but the left side's bevel has little dents from all those extra edge loops.



If you have extra cuts like this that aren't doing anything, you could remove some just to keep a cleaner mesh. But even so, there are always going to be extra cuts in your mesh, it's necessary to make modelling easier. So try playing around with limit options on your bevel modifier.

If you can't find a limit method that works only on the desired areas, you may have to apply a bevel manually by selecting the edges you want to bevel, and pressing ctrl+B. Be careful though, this is hard to undo if you decide later you don't like the bevels, and they make the mesh more complex, so it'll be more of a pain to work with. In fact, before applying bevels this way, you might want to make a complete copy of your mesh, and move it to a 'trash layer' as a backup, in case something goes wrong.